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How Learnable Grids Recover Fine Detail in Low Dimensions: A Neural Tangent Kernel Analysis of Multigrid Parametric Encodings
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Analyzes multigrid parametric encodings (MPE) versus Fourier feature encodings (FFE) through the neural tangent kernel, proving MPEs raise the NTK eigenvalue spectrum via their learnable grid, and validating on 2D image regression and 3D implicit surface regression.
Problem
Coordinate-based MLPs suffer from spectral bias and cannot learn high-frequency detail; although multigrid parametric encodings empirically outperform Fourier feature encodings, no NTK analysis existed for parametric encodings explaining why.
Approach
Derives the finite-width NTK for multigrid parametric encodings, shows via Weyl's inequality that the grid's learnable parameters add a positive semidefinite term that lower-bounds and raises the NTK eigenvalue spectrum, and isolates this gain to the learnable grid rather than the embedding space (unlike FFEs, which work through the embedding alone).
Key results
- MPE raises the minimum NTK eigenvalue by about 8 orders of magnitude over the baseline MLP and about 2 orders over the best FFE.
- On ImageNet tuned setting, MPE reaches 45.28 dB PSNR and 0.79 MS-SSIM versus 33.47 dB / 0.39 for FFE and 29.94 dB / 0.32 for baseline.
- Removing the learnable-grid contribution (MPE No Grid) leaves the spectrum barely above baseline, indicating the gain comes from the grid parameters.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use multigrid parametric encoding instead of no encoding | Coordinate-based MLP · ImageNet synonym sets | PSNR: 29.94 -> 45.28 | Helped |
| Use multigrid parametric encoding instead of Fourier feature encoding | Coordinate-based MLP · ImageNet synonym sets | PSNR: 33.47 -> 45.28 | Helped |
| Use multigrid parametric encoding instead of no encoding | Coordinate-based MLP · ImageNet synonym sets | MS-SSIM: 0.32 -> 0.79 | Helped |
| Use multigrid parametric encoding instead of Fourier feature encoding | Coordinate-based MLP · ImageNet synonym sets | MS-SSIM: 0.39 -> 0.79 | Helped |
| Use a fine MPE grid instead of no encoding | Coordinate-based MLP · Scaling image regression | PSNR: 20 -> 41.5 | Helped |
| Use a fine MPE grid instead of high-frequency Fourier features | Coordinate-based MLP · Scaling image regression | PSNR: 36.5 -> 41.5 | Helped |
| Increase Fourier feature frequency from low to mid | Coordinate-based MLP · Scaling image regression | PSNR: 25 -> 35 | Helped |
| Increase Fourier feature frequency from mid to high | Coordinate-based MLP · Scaling image regression | PSNR: 35 -> 36.5 | Helped |
| Use a fine rather than coarse MPE grid | Coordinate-based MLP · Scaling image regression | PSNR: 34.5 -> 41.5 | Helped |
| Use a fine MPE grid instead of no encoding | Coordinate-based MLP · Scaling image regression | MS-SSIM: 0.08 -> 0.73 | Helped |
| Use a fine MPE grid instead of high-frequency Fourier features | Coordinate-based MLP · Scaling image regression | MS-SSIM: 0.40 -> 0.73 | Helped |
| Increase Fourier feature frequency from low to high | Coordinate-based MLP · Scaling image regression | MS-SSIM: 0.20 -> 0.40 | Helped |
Surprising findings
- Without the learnable grid contribution, the MPE's embedding alone has little to no effect on the NTK spectrum compared to baseline.
- Coarse MPE crosses over the high-frequency FFE spectrum mid-plot, giving strong early training performance but lower final PSNR.
Limitations
- Analysis does not account for the influence of different activation functions (only ReLU considered in experiments).
- Computing the NTK for every application is too expensive; the finite-width NTK is not stable across all of training, breaking lazy-training assumptions.
- Evaluation is limited to 2D image regression and 3D occupancy on three meshes.
Experimental setup
Datasets: ImageNet Stanford 3D meshes (Armadillo, Buddha, Dragon)
Benchmarks: ImageNet 2D image regression (100 synonym sets) Stanford Armadillo/Buddha/Dragon implicit surface regression
Metrics: NTK eigenvalue spectrum PSNR MS-SSIM training loss
Baselines: Fourier feature encoding baseline MLP with no encoding
Builds on: neural tangent kernel Fourier feature encoding multigrid parametric encoding Weyl's inequality
Closest papers by meaning
- Grids Often Outperform Implicit Neural Representation at Compressing Dense Signals (NeurIPS 2025 · similarity 0.86)
- Learning Spatially Collaged Fourier Bases for Implicit Neural Representation (AAAI 2024 · similarity 0.86)
- Grounding and Enhancing Grid-based Models for Neural Fields (CVPR 2024 · similarity 0.85)
- Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domains (ICML 2024 · similarity 0.85)
- MIRE: Matched Implicit Neural Representations (CVPR 2025 · similarity 0.84)
- MetricGrids: Arbitrary Nonlinear Approximation with Elementary Metric Grids based Implicit Neural Representation (CVPR 2025 · similarity 0.84)
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